Transition formulae for ranks of abelian varieties

Abstract

Let $A_{/k}$ denote an abelian variety defined over a number field $k$ with good ordinary reduction at all primes above $p$, and let $K_{\infty }=\bigcup _{n\geq 1} K_n$ be a $p$-adic Lie extension of $k$ containing the cyclotomic $\mathbb{Z}_p$-extension. We use $\mathrm {K}-theory to find recurrence relations for the $\lambda$-invariant at each $\sigma$-component of the Selmer group over $K_{\infty }$, where $\sigma :G_k\rightarrow \mathrm{GL}(V)$. This provides upper bounds on the Mordell-Weil rank for $A(K_n)$ as $n\rightarrow \infty$ whenever $G_{\infty }=\mathrm {Gal}(K_{\infty}/k)$ has dimension at most $3$.